How to extrude curves without changing thickness?
Draw the desired squiggly curve, e.g. by Add | Curve | Bezier

Add a flat path object of the desired width, e.g. by Add | Curve | Path

Select the curve, click the object Data context, click the Bevel Object dropdown, and choose the flat path object, e.g. "NurbsPath" -- this will sweep the flat path object along the squiggly curve.

If you want a mesh object instead of a curve object, use Object | Convert To (Alt-C)